The blessing of a majority of the 11-member board could give Lee political cover to go back on his pledge not to run for a four-year term, a stance he took before he was appointed in January to serve out the remaining year of Mayor Gavin Newsom's term after Newsom was elected lieutenant governor.

The endorsement effort came as Lee on Wednesday refused to rule out running, saying, "There are interesting discussions going on."

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Those asking supervisors to back a Lee run appear short of garnering support from a majority of the board, where two members are mayoral candidates and two Lee allies said they don't want him to jump into the race.

Supervisor Sean Elsbernd, one of six supervisors who voted to appoint Lee as interim mayor in the crucial early round of voting when Lee garnered one more vote than Sheriff Michael Hennessey, said he had been repeatedly lobbied over the past week by leaders "across industries and, surprisingly, by people who I thought were lined up with other candidates."

"They have told me they are reaching out to all of the members of the board, or at least all of the members of the board who aren't running for mayor," Elsbernd said. "I've been clear ... that I wanted a (caretaker) mayor that would not be a candidate in November."

Supervisor Scott Wiener also said he had been asked to back a Lee candidacy. Wiener, who has endorsed City Attorney Dennis Herrera, his old boss, in the mayor's race, declined to say who made the pitch, other than to say it was not anyone from the mayor's staff or Lee's prominent backers, including former Mayor Willie Brown or Chinatown power broker Rose Pak.

Other supervisors, including Ross Mirkarimi, said they'd had informal discussions with supporters of a Lee candidacy but weren't asked to publicly back him.

The current board, though, is largely the wrong target if Lee wants to be released from his promise not to run, Elsbernd said.

"If the intent of this is to relieve the mayor of his pledge to the board, it's not necessarily this board that should be consulted, but the six members of the board who last January truly elected this mayor," Elsbernd said. "If not for the six of us, Mike Hennessey would be mayor right now."

The six - Elsbernd, board President David Chiu, Supervisor Carmen Chu and former Supervisors Bevan Dufty, Michela Alioto-Pier and Sophie Maxwell - cast the critical votes that were essentially ratified a few days later when a newly sworn-in board voted 11-0 to appoint Lee.

Only one of those original six - Carmen Chu - voiced support for a Lee candidacy. Three of the others - Chiu, Dufty and Alioto-Pier - are already candidates

"I think that the possibility was always out there," said Chu, who said she never asked Lee to promise not to run before she voted for him. "I certainly think he owes it to himself and the city to think about it, to consider it."

Others see a possible Lee run as a betrayal.

"I certainly would feel like he did not keep his word," Maxwell said. "That would be very troubling to me, that he did not keep his word, because based on his word, I voted for him."

Chiu and Dufty both said they believe Lee to be a man of his word and don't expect him to run.

If he does, Maxwell, a staunch Herrera backer, said Lee won't get her vote again - even under the city's ranked-choice voting system that allows voters to pick up to three candidates.

"Would Ed Lee be one of my three people?" Maxwell said. "No, because he said he would not run."